Divide using synthetic division $$ ( 9x^{3}+8x^{2}-17x-10 ) \div ( 2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&9&8&-17&-10\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{9}&\color{blue}{8}&\color{blue}{-17}&\color{orangered}{-10} \end{array} $$The solution is:
$$ \frac{ 9x^{3}+8x^{2}-17x-10 }{ x } = \color{blue}{9x^{2}+8x-17} \color{red}{~-~} \frac{ \color{red}{ 10 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&9&8&-17&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ 9 }&8&-17&-10\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 9 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&9&8&-17&-10\\& & \color{blue}{0} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 0 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}0&9&\color{orangered}{ 8 }&-17&-10\\& & \color{orangered}{0} & & \\ \hline &9&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 8 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&9&8&-17&-10\\& & 0& \color{blue}{0} & \\ \hline &9&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 0 } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}0&9&8&\color{orangered}{ -17 }&-10\\& & 0& \color{orangered}{0} & \\ \hline &9&8&\color{orangered}{-17}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&9&8&-17&-10\\& & 0& 0& \color{blue}{0} \\ \hline &9&8&\color{blue}{-17}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 0 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}0&9&8&-17&\color{orangered}{ -10 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{9}&\color{blue}{8}&\color{blue}{-17}&\color{orangered}{-10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{2}+8x-17 } $ with a remainder of $ \color{red}{ -10 } $.